Question: $ 0.\overline{93} \div -1.\overline{1} = {?} $
First convert the repeating decimals to fractions. $\begin{align*} 100x &= 93.9393...\\ x &= 0.9393...\end{align*} $ $\begin{align*} 99x &= 93 \\ x &= \dfrac{93}{99}\end{align*} $ $\begin{align*} 10y &= -11.1112...\\ y &= -1.1112...\end{align*} $ $\begin{align*} 9y &= -10 \\ y &= -\dfrac{10}{9}\end{align*} $ So, the problem becomes: $ \dfrac{93}{99} \div -\dfrac{10}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{93}{99} \times -\dfrac{9}{10} = {?} $ $ \phantom{\dfrac{93}{99} \times -\dfrac{10}{9}} = \dfrac{93 \times 9}{99 \times -10} $ $ \phantom{\dfrac{93}{99} \times -\dfrac{10}{9}} = \dfrac{93 \times \cancel{9}} {\cancel{99}11 \times -10} $ $ \phantom{\dfrac{93}{99} \times -\dfrac{10}{9}} = -\dfrac{93}{110} $